We now turn to the proof of Claim REF .
This type of construction is usually tedious because of the general form of \(\mathcal {F}\) , but may be performed fairly explicitly.
We warn the reader of the difficulties arising from the potential bottlenecks of \(\mathcal {F}\)  and the fact that the RSW result (REF ) is not valid with arbitrary boundary conditions in arbitrary quads when \(q=4\) .
Nevertheless, with sufficient care, the domains \(D_1,\dots ,D_4\)  may be constructed as unions of small squares \(\Lambda _{c \rho n}(x)\)  with \(x \in c\rho \mathbb {Z}^2\)  and \(c > 0\)  a small constant independent of \(\rho \) .
We propose below an alternative, more innovative construction of \(D_1,\dots , D_4\) .
We will use the following result from [1]} (see Remark 4.2 to be precise).
